Problem

Source: ELMO Shortlist 2024/C2

Tags: Elmo, combinatorics



Let $n$ be a fixed positive integer. Ben is playing a computer game. The computer picks a tree $T$ such that no vertex of $T$ has degree $2$ and such that $T$ has exactly $n$ leaves, labeled $v_1,\ldots, v_n$. The computer then puts an integer weight on each edge of $T$, and shows Ben neither the tree $T$ nor the weights. Ben can ask queries by specifying two integers $1\leq i < j \leq n$, and the computer will return the sum of the weights on the path from $v_i$ to $v_j$. At any point, Ben can guess whether the tree's weights are all zero. He wins the game if he is correct, and loses if he is incorrect. (a) Show that if Ben asks all $\binom n2$ possible queries, then he can guarantee victory. (b) Does Ben have a strategy to guarantee victory in less than $\binom n2$ queries? Brandon Wang